Absolutely! You've hit on a super important concept in calculus. The General Power Rule combined with the Chain Rule is like the dynamic duo for differentiating functions that look like 'something raised to a power'. Let's break it down so it clicks for you! โจ
What is the General Power Rule with the Chain Rule? ๐ค
At its heart, this rule helps us find the derivative of a function that has an outer function (something raised to a power) and an inner function (the 'something' inside the parentheses). Think of it as peeling an onion: you differentiate the outer layer first, then multiply by the derivative of the inner layer.
The formula for the derivative of a function like \( [g(x)]^n \) with respect to \( x \) is:
\[ \frac{d}{dx} [g(x)]^n = n[g(x)]^{n-1} \cdot g'(x) \]
Let's dissect this a bit:
- \( \mathbf{g(x)} \) is your inner function (the base of the power).
- \( \mathbf{n} \) is the exponent (the power itself).
- \( \mathbf{n[g(x)]^{n-1}} \) is the result of applying the standard Power Rule to the 'outer' function. You bring the power down and subtract one from the exponent, just like you would with \( x^n \).
- \( \mathbf{g'(x)} \) is the derivative of your inner function. This is the 'chain' part โ you multiply by the derivative of what was inside the parentheses.
Let's Break It Down: Step-by-Step ๐
When you encounter a function like \( f(x) = (\text{something})^n \), follow these steps to find its derivative \( f'(x) \):
- Identify the inner function \( g(x) \) and the power \( n \).
- Apply the Power Rule to the 'outer' structure: Bring the power \( n \) down to the front and subtract 1 from the exponent. Keep the inner function \( g(x) \) exactly as it is inside the parentheses. So you'll have \( n[g(x)]^{n-1} \).
- Multiply by the derivative of the inner function: Find \( g'(x) \) and multiply your result from Step 2 by it.
Example Time! โ๏ธ
Let's find the derivative of \( f(x) = (3x^2 + 5)^4 \).
- Step 1: Identify \( g(x) \) and \( n \).
- Inner function: \( g(x) = 3x^2 + 5 \)
- Power: \( n = 4 \)
- Step 2: Apply the Power Rule to the outer structure.
- Bring down the 4 and reduce the exponent by 1: \( 4(3x^2 + 5)^{4-1} = 4(3x^2 + 5)^3 \)
- Step 3: Find the derivative of the inner function \( g'(x) \) and multiply.
- Derivative of \( g(x) = 3x^2 + 5 \) is \( g'(x) = 6x \).
- Now, multiply: \( f'(x) = 4(3x^2 + 5)^3 \cdot (6x) \)
Simplifying the expression, we get: \( f'(x) = 24x(3x^2 + 5)^3 \)
Pro Tip! ๐ก Always think of it as \("derivative of the outside, leaving the inside alone, multiplied by the derivative of the inside."\) This mental shortcut can save you from errors! Keep practicing, and it will become second nature. You've got this! ๐ช
